Speed and Velocity Topic sentence: Even though there is one speed there are many different velocities during circular motion. First Paragraph: The average speed of something traveling in a circular path is simply the distance, 2piR, over time; however, even though the path of the object is constant, the velocity is not. The object is still changing direction, constantly, and this forms a circle. These velocities are all tangent to the circle.
Acceleration Topic sentence: Even though the speed is constant it doesn't mean there isn't acceleration! First Paragraph: Notice how the direction of the object changes even though the speed doesn't, how the velocity is not constant because it is a vector. This change in direction means there must be acceleration present.
The Centripetal Force Requirement Topic sentence: Smoke means fire and acceleration means force. First Paragraph: This acceleration has to be caused by something right? Well some force, like normal force, tension, gravity, ect, has to act as a centripetal force and constantly push or pull the object in a circular path. If there were no centripetal force then the object would just go straight because of it's inertia. These forces always point toward the center of the circle, never otherwise!
The Forbidden F-Word Topic sentence: Centrifugal force doesn't exist. Plain and simple. First Paragraph: Centrifugal means on the outside of the circle and they're simply aren't any forces outside the circle. The 'force' that would make someone on an merry go round gone haywire feel like they're being pulled to the outside of the circle is actually the centripetal force pushing them in, not some magic force pulling them out.
The Mathematics of Circular Motion Topic sentence: Everything boils down to newton's laws (maybe that's why they call it Newtonian physics) First Paragraph: The mathematics behind how the following equation is derived from newton's three laws are complicated and long winded, trust me I checked. Long story short the acceleration that on object traveling in circular motion is equal to V^2/R. Take my word for it. You can then use this equation for all sorts of fun and games, if that's what your call fun and games.
Homework Circular Motion Unit Lesson 2
Newton's Second Law - Revisited Topic sentence: Not going anywhere. First Paragraph: F = ma can also be applied in the context Fcentripetal = m * Acentripetal. This means that in any uniform circular motion situation Fc can be calculated by knowing Ac and M. As for which force(s) it is only a simple free body diagram is required to determine that.
AmusementPark Physics Topic sentence: The physics makes it fun! First Paragraph: In any given amusement park there are many different types of physics at work. Circular motion physics take part in things like loops, dip hills and banked turns. Something interesting to observe is that in a full circular loop, assuming the speed is constant and Fc is constant, since Fc must always point towards the circle, when you're at the bottom of the loop normal force has to overcome gravity AND provide for Fc while at the top only gravity is on your side and takes care of Fc so normal force doesn't have to. You might also notice that there is more force on smaller or quicker turns, this is all because of the math above.
Athletics Topic sentence: Physics physics everywhere! First Paragraph: Physics is present all over the universe, that's why it's physics. Even athletes making a turn (even a varying turn, hello calc!) are subject to the laws of physics. Everything that applies above still applies.
Homework Circular Motion Unit Lesson 3
Gravity is more then a Name Topic sentence: Gravity, more clingy then your ex First Paragraph: The acceleration of gravity is the acceleration that something under the force of gravity, Fgrav, experiences on earth. However this is not the same as the force of gravity itself, merely evidence of it's existence.
The Apple, the Moon, and the inverse Square Law Topic sentence: Flavors of gravity First Paragraph: Johannes Kepler used math and science to prove that the planets moved in a elliptical path, but couldn't explain why. Newton proposed the theory of universal gravitation, that gravity spread out over everywhere but wasn't as strong the further away you got. Comparing the acceleration of an apple on the earth's surface to the moon, not on the earth's surface, and found that the ratio of Amoon to Aapple was the same as the inverse of the ratio of DistanceMoon^2 to DistanceApple^2. Thus the inverse square law was born.
Newton's Law of universal Gravitation Topic sentence: Weight changes the flavor First Paragraph: Not only is distance a factor in the strength of Fgrav but also the masses of both objects in question. Also both the apple and earth were attracted to one another, although the apple was just a lot more moved compared to the earth (Hey! sorta like your ex). He discovered that Fgrav was proportional to the product of the masses divided by the distance separating them. Since that number above is linearly proportional the only thing it takes to make it equal instead is a constant, this is the universeal gravatational constant G, G = 6.673 x 10-11 N m2/kg2
Cavendish and the Value of G Topic sentence: Newton wasn't alone. First Paragraph: Even though Newton theorized the existence of G it was never experimentally determined until 1798. Lord Henry Canvish he used a torsion balance that would rotate, but not revolve, a different amount depending on the force causing it to rotate. He put two weights at either end of the apparatus and two more weights in front of them them measured how much the balance rotated because of the gravity pulling the two objects together. Thus science was performed.
The Value of g Topic sentence: How we get g from G First Paragraph: If you take the equation of universal gravitation Fgrav = M1 * M2 * G / D^2 and substitute in Mearth for M1, call D pretty much the same relative to the surface of the earth (the changes between the ceiling and the floor doesn't matter that much) and set the whole thing equal to F = M2 A2 you get (Mearth * G * M1) / D^2 = M2 A2, the two M2s cancel out and the rest are now numbers which roughly simplify to g, 9.8 m/s. If you get really far from the center of the earth then g changes but that's another topic.
Homework The Clockwork Universe
Finding the Center Topic sentence: Copernicus sparked the debate about what the true center of the universe was but that idea took a while to grow in to a fire. First Paragraph: Copernicus was the first to propose a heliocentric model of the universe, were everything orbited the sun, not the earth, but his ideas didn't catch on. Galalio at first supported his ideas but was "convinced" by the church to renounce them. Kepler then worked off Copernicus's theory and proposed the planets moved in ellipses but did not know why.
The Apex of the Blaze Topic sentence: Newton was the right arson in the right dry forest, I'm sure that can't be misinterpreted. First Paragraph: Newton was alive in the time when there was both a demand for an explanation of Kepler's laws and the geometrical tools to find them. He focused not on motion but rather on change in constant motion, acceleration. He developed theories that wouldn't be challenged until Einstein, publishing his universal theory of gravitation.
...It was Always Burning Since the World was Turning Topic sentence: Newton's one law for universal gravitation, combined with his general laws of motion, made it possible for him to mathematically predict how a single planet would move around the sun. First Paragraph: Aside form this being a great scientific feat it also raised some philosophical questions. As more and more science was developed to predict, well everything, people began to wonder about determinism, the idea that there was only one possible outcome, overriding the idea of free will. This was a controversial topic and is still not settled today.
Homework Circular Motion Unit Lesson 4
Kepler's Laws Further Explained Topic sentence: Kepler's three laws are not only accurate but can be proven so. First Paragraph: Kepler's first law simply states that the planets will orbit following an ellipse with the sun at one of the foci, for more information regarding ellipses review conic sections. Kepler's second law, the law of equal areas states that the area carved out by a line from the center of the planet to the center of the sun will be the same for any given time period as the area carved out in that time period at any other location. In other words, if the ellipse was a pie, sliced from planet to sun, no matter where you start if you wait the same time before saying "stop" (when the person cutting the pie cuts the pie) you will always get the same size piece. The third law is the law or harmonies, it states that there is a constant ratio of period squared over radius cubed for any planet that orbits the sun. So if you know the radius and period on any given planet and the radius or period of any other planet you can calculate all your missing information.
Satellitely Familiar Topic sentence: Before reading this brush up on constant circular motion and sit back in awe at the similar (actually the same) math. First Paragraph: A satellite is anything that orbits anything else, natural or not (eg: both the moon and GPS satellites are satellites). While in actuality satellites do not go in perfect circles, unless the ellipse they follow happens to have an eccentricity of 0 (which will happen more often then not in these examples), when talking about averages the same math used in circular motion can be applied to satellite motion by simply remembering that Fgrav exists.
Completely New and Never-Before-Seen Material. Topic sentence: Remember the math from above, don't forget it. First Paragraph: Just reread the above paragraph and remember that Fgrav is often the (only) force pulling toward the planet. Nearly always Fgrav (G*m1*m2/D^2) just equals ma (m*V^2/R^2).
Going to Space, the New Diet Craze Topic sentence: In Earth's orbit nothing about a person's mass changes however they fell weightless none the less. First Paragraph: Apparent weight (what a scale would read) only measured the amount of normal force an object is exerting on a plate. On earth there is a constant downward acceleration of -9.8 m/s/s that makes weight seem real. In space if the astronaut is acceleration at the same rate as the spacecraft (both around the earth) relative to each other there is no acceleration and therefor no force. Therefor there is no apparent weight.
Table of Contents
Homework Circular Motion Unit Lesson 1
Speed and Velocity
Topic sentence: Even though there is one speed there are many different velocities during circular motion.
First Paragraph: The average speed of something traveling in a circular path is simply the distance, 2piR, over time; however, even though the path of the object is constant, the velocity is not. The object is still changing direction, constantly, and this forms a circle. These velocities are all tangent to the circle.
Acceleration
Topic sentence: Even though the speed is constant it doesn't mean there isn't acceleration!
First Paragraph: Notice how the direction of the object changes even though the speed doesn't, how the velocity is not constant because it is a vector. This change in direction means there must be acceleration present.
The Centripetal Force Requirement
Topic sentence: Smoke means fire and acceleration means force.
First Paragraph: This acceleration has to be caused by something right? Well some force, like normal force, tension, gravity, ect, has to act as a centripetal force and constantly push or pull the object in a circular path. If there were no centripetal force then the object would just go straight because of it's inertia. These forces always point toward the center of the circle, never otherwise!
The Forbidden F-Word
Topic sentence: Centrifugal force doesn't exist. Plain and simple.
First Paragraph: Centrifugal means on the outside of the circle and they're simply aren't any forces outside the circle. The 'force' that would make someone on an merry go round gone haywire feel like they're being pulled to the outside of the circle is actually the centripetal force pushing them in, not some magic force pulling them out.
The Mathematics of Circular Motion
Topic sentence: Everything boils down to newton's laws (maybe that's why they call it Newtonian physics)
First Paragraph: The mathematics behind how the following equation is derived from newton's three laws are complicated and long winded, trust me I checked. Long story short the acceleration that on object traveling in circular motion is equal to V^2/R. Take my word for it. You can then use this equation for all sorts of fun and games, if that's what your call fun and games.
Homework Circular Motion Unit Lesson 2
Newton's Second Law - RevisitedTopic sentence: Not going anywhere.
First Paragraph: F = ma can also be applied in the context Fcentripetal = m * Acentripetal. This means that in any uniform circular motion situation Fc can be calculated by knowing Ac and M. As for which force(s) it is only a simple free body diagram is required to determine that.
Amusement Park Physics
Topic sentence: The physics makes it fun!
First Paragraph: In any given amusement park there are many different types of physics at work. Circular motion physics take part in things like loops, dip hills and banked turns. Something interesting to observe is that in a full circular loop, assuming the speed is constant and Fc is constant, since Fc must always point towards the circle, when you're at the bottom of the loop normal force has to overcome gravity AND provide for Fc while at the top only gravity is on your side and takes care of Fc so normal force doesn't have to. You might also notice that there is more force on smaller or quicker turns, this is all because of the math above.
Athletics
Topic sentence: Physics physics everywhere!
First Paragraph: Physics is present all over the universe, that's why it's physics. Even athletes making a turn (even a varying turn, hello calc!) are subject to the laws of physics. Everything that applies above still applies.
Homework Circular Motion Unit Lesson 3
Gravity is more then a Name
Topic sentence: Gravity, more clingy then your ex
First Paragraph: The acceleration of gravity is the acceleration that something under the force of gravity, Fgrav, experiences on earth. However this is not the same as the force of gravity itself, merely evidence of it's existence.
The Apple, the Moon, and the inverse Square Law
Topic sentence: Flavors of gravity
First Paragraph: Johannes Kepler used math and science to prove that the planets moved in a elliptical path, but couldn't explain why. Newton proposed the theory of universal gravitation, that gravity spread out over everywhere but wasn't as strong the further away you got. Comparing the acceleration of an apple on the earth's surface to the moon, not on the earth's surface, and found that the ratio of Amoon to Aapple was the same as the inverse of the ratio of DistanceMoon^2 to DistanceApple^2. Thus the inverse square law was born.
Newton's Law of universal Gravitation
Topic sentence: Weight changes the flavor
First Paragraph: Not only is distance a factor in the strength of Fgrav but also the masses of both objects in question. Also both the apple and earth were attracted to one another, although the apple was just a lot more moved compared to the earth (Hey! sorta like your ex). He discovered that Fgrav was proportional to the product of the masses divided by the distance separating them. Since that number above is linearly proportional the only thing it takes to make it equal instead is a constant, this is the universeal gravatational constant G, G = 6.673 x 10-11 N m2/kg2
Cavendish and the Value of G
Topic sentence: Newton wasn't alone.
First Paragraph: Even though Newton theorized the existence of G it was never experimentally determined until 1798. Lord Henry Canvish he used a torsion balance that would rotate, but not revolve, a different amount depending on the force causing it to rotate. He put two weights at either end of the apparatus and two more weights in front of them them measured how much the balance rotated because of the gravity pulling the two objects together. Thus science was performed.
The Value of g
Topic sentence: How we get g from G
First Paragraph: If you take the equation of universal gravitation Fgrav = M1 * M2 * G / D^2 and substitute in Mearth for M1, call D pretty much the same relative to the surface of the earth (the changes between the ceiling and the floor doesn't matter that much) and set the whole thing equal to F = M2 A2 you get (Mearth * G * M1) / D^2 = M2 A2, the two M2s cancel out and the rest are now numbers which roughly simplify to g, 9.8 m/s. If you get really far from the center of the earth then g changes but that's another topic.
Homework The Clockwork Universe
Finding the Center
Topic sentence: Copernicus sparked the debate about what the true center of the universe was but that idea took a while to grow in to a fire.
First Paragraph: Copernicus was the first to propose a heliocentric model of the universe, were everything orbited the sun, not the earth, but his ideas didn't catch on. Galalio at first supported his ideas but was "convinced" by the church to renounce them. Kepler then worked off Copernicus's theory and proposed the planets moved in ellipses but did not know why.
Growing the Flame
Topic sentence: Even though Kepler could not explain why he was right new advances in mathematics garnered support for his ideas.
First Paragraph: With René Descartes's advanced in relating geometry to algebra geometry was in turn developed more. Since scientists could now take a more mathematical approach to geometry they could further extend Kepler's ideas.
The Apex of the Blaze
Topic sentence: Newton was the right arson in the right dry forest, I'm sure that can't be misinterpreted.
First Paragraph: Newton was alive in the time when there was both a demand for an explanation of Kepler's laws and the geometrical tools to find them. He focused not on motion but rather on change in constant motion, acceleration. He developed theories that wouldn't be challenged until Einstein, publishing his universal theory of gravitation.
...It was Always Burning Since the World was Turning
Topic sentence: Newton's one law for universal gravitation, combined with his general laws of motion, made it possible for him to mathematically predict how a single planet would move around the sun.
First Paragraph: Aside form this being a great scientific feat it also raised some philosophical questions. As more and more science was developed to predict, well everything, people began to wonder about determinism, the idea that there was only one possible outcome, overriding the idea of free will. This was a controversial topic and is still not settled today.
Homework Circular Motion Unit Lesson 4
Kepler's Laws Further Explained
Topic sentence: Kepler's three laws are not only accurate but can be proven so.
First Paragraph: Kepler's first law simply states that the planets will orbit following an ellipse with the sun at one of the foci, for more information regarding ellipses review conic sections. Kepler's second law, the law of equal areas states that the area carved out by a line from the center of the planet to the center of the sun will be the same for any given time period as the area carved out in that time period at any other location. In other words, if the ellipse was a pie, sliced from planet to sun, no matter where you start if you wait the same time before saying "stop" (when the person cutting the pie cuts the pie) you will always get the same size piece. The third law is the law or harmonies, it states that there is a constant ratio of period squared over radius cubed for any planet that orbits the sun. So if you know the radius and period on any given planet and the radius or period of any other planet you can calculate all your missing information.
Satellitely Familiar
Topic sentence: Before reading this brush up on constant circular motion and sit back in awe at the similar (actually the same) math.
First Paragraph: A satellite is anything that orbits anything else, natural or not (eg: both the moon and GPS satellites are satellites). While in actuality satellites do not go in perfect circles, unless the ellipse they follow happens to have an eccentricity of 0 (which will happen more often then not in these examples), when talking about averages the same math used in circular motion can be applied to satellite motion by simply remembering that Fgrav exists.
Completely New and Never-Before-Seen Material.
Topic sentence: Remember the math from above, don't forget it.
First Paragraph: Just reread the above paragraph and remember that Fgrav is often the (only) force pulling toward the planet. Nearly always Fgrav (G*m1*m2/D^2) just equals ma (m*V^2/R^2).
Going to Space, the New Diet Craze
Topic sentence: In Earth's orbit nothing about a person's mass changes however they fell weightless none the less.
First Paragraph: Apparent weight (what a scale would read) only measured the amount of normal force an object is exerting on a plate. On earth there is a constant downward acceleration of -9.8 m/s/s that makes weight seem real. In space if the astronaut is acceleration at the same rate as the spacecraft (both around the earth) relative to each other there is no acceleration and therefor no force. Therefor there is no apparent weight.
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